Nonlinear-frequency-packing nonlinear frequency division multiplexing transmission

Xulun Zhang; Peng Sun; Lixia Xi; Zibo Zheng; Shucheng Du; Jiacheng Wei; Yue Wu; Xiaoguang Zhang

doi:10.1364/OE.390293

1. Introduction

Kerr nonlinearity has been considered as a major factor limiting the transmission capacity in high speed and long-haul optical fiber communications [1]. In recent years, nonlinear frequency division multiplexing (NFDM) transmission based on the nonlinear Fourier transform (NFT), a revolutionary scheme, has obtained great attention due to its natural immunity to Kerr nonlinearity impairment [2–6]. The NFT (also called the inverse scattering transform) provides a theoretical solution to the nonlinear Schrödinger equation (NLSE), which can be used for describing the propagation of optical signal along single mode fiber (SMF) channel [2]. Enabled by NFT, the information can be modulated on (demodulated from) the nonlinear spectrum, which is composed of discrete spectrum (soliton component) and continuous spectrum (dispersive component). The benefit of this transform is to make sure the signal avoids any crosstalk when goes through NLSE channel, rather than severe distortion if it is modulated on time or linear frequency domain [2–6]. The transfer function of NLSE in nonlinear Fourier domain is quite simple and we only need one tap equalization to remove propagation effects [7,8]. A number of proof-of-concept experiments for NFDM transmission have been demonstrated in last few years [9].

However, practical application for NFDM still faces many challenges, such as the signal-noise interaction in the nonlinear Fourier domain, the mismatch between the ideal channel model of NFDM and practical fiber link with lumped amplification, and the inaccuracy of the NFT algorithm at high launch power [10], etc. All those aspects inhibit the increasing of spectral efficiency (SE) and transmission distance. So far, a lot of impressive efforts and contributions have been made for data transmission by exploiting NFDM. Discrete and continuous spectrum have been exploring separately. In the discrete spectrum modulation cases, despite a lot of effort and work have been done [10–15], the SE of the currently demonstrated NFDM with only discrete spectrum modulation is quite low (less than 1 bit/s/Hz) [10] due to the soliton essence of discrete eigenvalues. In the continuous spectrum modulation cases, to fulﬁll the vanishing boundary conditions and avoid burst interaction during propagation, a guard interval (GI) between the two adjacent burst signals is necessary, which severely limits the SE [10]. The pre-dispersion compensation (PDC) technique has been proposed, which can halve the GI [16,17]. A modified continuous spectrum modulation scheme, b-coefficient modulation, has been proposed to generate a time-limited and compact signal [18–23]. This scheme reduces the impact of tail truncation and noise, but has marginal impact on the SE [10]. In addition, simply increasing the number of nonlinear subcarriers or QAM symbols (in nonlinear Fourier domain) does not improve the SE because of a detrimental signal–noise interaction [17]. At present, there is still a lack of a novel modulation scheme that can reduce the impact of noise and increase the SE significantly. The NFDM system has been extended from the single polarization to dual polarization transmission [13,24]. An experiment with the SE of 4 bit/s/Hz has been demonstrated through modulating 392 nonlinear subcarriers in dual polarization b-modulation system [23]. The NFDM system with full spectrum modulation has also been explored, but the crosstalk between continuous spectrum and discrete spectrum become significant in practical link conditions [8,10,25]. The first experimental work about dual polarization and full spectrum modulation has been reported. It demonstrates the modulation of all the degree of freedom in nonlinear Fourier domain over SMF [26]. However, to our best knowledge, the SE of currently demonstrated NFDM system are still not as good as traditional linear system and not satisfied for the capacity demand from communication network.

In this work, we reconsider the strategy of increasing the SE from a brand-new perspective, and design a new encoding scheme, in which the subcarriers are packed as close as possible to increase the SE. The idea of this proposed scheme originates from the concept of faster than Nyquist (FTN), which was proposed by Mazo in 1970s [27]. In FTN system, the adjacent pulse can be packed tighter than Nyquist limit (time packing). The orthogonality of the packed pulse is destroyed and the inter-symbol interference is introduced. But employing the optimal detector within the Mazo limit, there is no performance degradation and more information can be transmitted. In addition, the concept of FTN has been extended to frequency domain (frequency packing). The subcarriers in multi-subcarrier system are packed tighter, which is called spectrally efficient frequency division multiplexing (SEFDM) [28,29]. Compared to the orthogonal frequency division multiplexing system, the spacing of the subcarriers is squeezed and more subcarriers can be packed in the same linear bandwidth. And the time–frequency packing has also been studied [29,30]. In NFDM system, the information is modulated in the nonlinear Fourier domain (NFD). Therefore, we extend the core idea of FTN packing from the time (or frequency) domain to NFD and name such scheme nonlinear frequency packing (NFP) NFDM. To the best of our knowledge, this is the first time for proposing NFP-NFDM transmission scheme to further increase the SE.

The remainder of this paper is organized as follows. We firstly review the fiber channel model and fundamental principle of NFT in Section 2. Then, in Section 3, the current modulation scheme and the strategy of increasing the SE are analyzed. Afterwards, we propose NFP-NFDM modulation scheme in Section 4. Section 5 describes the DSP processing flow at the transmitter and receiver side. We compare the performance of NFDM and NFP-NFDM. The upper bound of their corresponding SE is analyzed and some comments and perspective about NFP-NFDM are presented in Section 6. At last, the conclusions of this work are given in section 7.

2. Channel model and nonlinear Fourier transform

In this work, for the simplicity, we only consider the single polarization transmission scenario. The proposed scheme can also be extended to the dual polarization transmission system. In this section, we briefly review the channel model of optical fiber communication, and the basic principle of NFT as well as its fundamental properties.

The propagation of optical signal $E(z,t)$ along the SMF channel can be modeled as NLSE [31].

(1)$$i\frac{{\partial E}}{{\partial z}} - \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}E}}{{\partial {t^2}}} + \gamma {|E |^2}E ={-} i\frac{\alpha }{2}E,$$

where z (km) is the propagation distance, t (ps) stands for the retarded time co-moving with the group velocity of the signal, ${\beta _2}$ (ps²km^-1) is the group velocity dispersion coefficient, $\gamma$ (W^-1km^-1) is the Kerr nonlinearity parameter and $\alpha$(km^-1) is the attenuation coefficient. The NFT only applies to the lossless NLSE scenario [2]. However, for a practical fiber channel, losses are inevitable and the Erbium-doped fiber amplifier (EDFA) is used to compensate losses. To mitigate the mismatch between the lossless NLSE and practical fiber channel model, a lossless path-averaged (LPA) NLSE model is employed [32]. By introducing the new variable $\bar{E} = {e^{ - ({\alpha /2} )z}}E$, the LPA NLSE is obtained

(2)$$i\frac{{\partial \bar{E}}}{{\partial z}} - \frac{{{\beta _2}}}{2}\frac{{{\partial ^2}\bar{E}}}{{\partial {t^2}}} + {\gamma _1}{|{\bar{E}} |^2}\bar{E} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\gamma _1} = \gamma (1 - {e^{ - \alpha L}})/(\alpha L),$$

where L is the span length. The LPA NLSE can be written as a normalized form through introducing the normalized parameters

(3)$$\tau = t/{T_S},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} l = z/{Z_S},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {Z_S} = 2{T_S}^2/|{{\beta_2}} |,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} q = \bar{E}/{A_S},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {A_S} = \sqrt {2/({\gamma _1}{Z_S})} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa ={-} {\mathop{\rm sgn}} ({\beta _2}),$$

where ${T_S}$ is a normalized time parameter and we set ${T_S} = 1ns$. Then the normalized NLSE can be simplified to

(4)$$i\frac{{\partial q}}{{\partial l}} + \kappa \frac{{{\partial ^2}q}}{{\partial {\tau ^2}}} + 2{|q |^2}q = 0,$$

In this work, we choose widely used standard single mode fiber (SSMF) as the transmission media, and its group velocity dispersion coefficient satisfies ${\beta _2} < 0$ ($\kappa ={+} 1$).

The NFT is a powerful mathematical tool to solve the integrable partial differential equations, such as the lossless NLSE [2,33]. The nonlinear spectrum of a time domain pulse $q(\tau )$ (its boundary tends to zero) can be calculated by solving the Zakharov-Shabat problem with an initial condition [2]

(5)$$\frac{\partial }{{\partial \tau }}\upsilon = \left( {\begin{array}{{cc}} { - j\lambda }&{q(\tau )}\\ { - {q^{\ast }}(\tau )}&{j\lambda } \end{array}} \right)\upsilon {\kern 1pt} {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop {\lim }\limits_{\tau \to - \infty } \upsilon (\tau ,\lambda ) = \left( {\begin{array}{{c}} 1\\ 0 \end{array}} \right){e^{ - j\lambda \tau }},$$

where $\upsilon = {({{\upsilon_1},{\upsilon_2}} )^T}$ is the eigenvector corresponding to the eigenvalue λ. The scattering coefficients $a(\lambda ) = \mathop {\lim }\limits_{\tau \to \infty } {\upsilon _1}(\tau ){e^{j\lambda \tau }}$ and $b(\lambda ) = \mathop {\lim }\limits_{\tau \to \infty } {\upsilon _2}(\tau ){e^{ - j\lambda \tau }}$. For continuous spectrum system, the NFT of $q(\tau )$ is defined as $\hat{q}(\lambda ) = b(\lambda )/a(\lambda ),{\kern 1pt} {\kern 1pt} \lambda \in {\mathbb R}.$ The evolution of the scattering coefficients and continuous spectrum along the distance satisfy the following rules:

(6)$$a(\lambda ,l) = a(\lambda ,0){\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b(\lambda ,l) = b(\lambda ,0)\exp (4j{\lambda ^2}l),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \hat{q}(\lambda ,l) = \hat{q}(\lambda ,0)\exp (4j{\lambda ^2}l),$$

where l is the normalized distance. In the continuous spectrum system (assuming that there is no discrete spectrum), the energy of time domain signal, its corresponding continuous spectrum and scattering coefficient satisfy

(7)$$\int_{ - \infty }^\infty {{{|{q(t)} |}^2}dt} = \frac{1}{\pi }\int_{ - \infty }^\infty {\log ({1 + {{|{\hat{q}(\lambda )} |}^2}} )} d\lambda ={-} \frac{1}{\pi }\int_{ - \infty }^\infty {\log ({1 - {{|{b(\lambda )} |}^2}} )} d\lambda .$$

3. NFDM transmissions with b-modulation and the strategy of increasing SE

Some studies have shown that modulation on the scattering coefficient $b(\lambda )$ behaves better transmission performance than that on the continuous spectrum $\hat{q}(\lambda )$, because it can generate more compact time domain signal and reduce the signal-noise interaction [18–23]. However, directly b-modulation is limited by energy barrier of $b(\lambda )$ [18,19,21]. The b-modulation plus “U modulation” scheme can solve the problem, but the full control of the duration of time domain signal is lost [20,22–23]. In this work, we adopt the scheme of b-modulation plus “U modulation”. First, the initial signal is modulated as

(8)$$U(\lambda ) = A\sum\limits_{k ={-} (N/2)}^{N/2 - 1} {{s_k}\frac{{\sin (\lambda {T_0}/{T_S} + k\pi )}}{{\lambda {T_0}/{T_S} + k\pi }}} ,$$

where N is the quantity of nonlinear subcarriers, ${s_k}$ denotes symbol drawn from QAM constellation, ${T_0}$ is the useful block time and A is used to control the energy of signal. Then the b coefficient is defined as

(9)$$b(\lambda ) = \sqrt {1 - \exp ( - {{|{U(\lambda )} |}^2})} \frac{{U(\lambda )}}{{|{U(\lambda )} |}}.$$

The NFDM time domain signal is generated by the inverse nonlinear Fourier transform (INFT).

As described in [34], in the given bandwidth B and transmission distance L, the GI can be estimated approximately as

(10)$${T_{GI}} = 2\pi B{\beta _2}L.$$

The useful block time of signal can be estimated as ${T_\textrm{0}} = N/B$. For the single polarization transmission system, the upper bound of the normalized SE (symbol/s/Hz, free from the order of modulation format) for the above-mentioned NFDM transmission scheme can be expressed as

(11)$$S{E_{NFDM}} = \frac{{{T_0}}}{{{T_0} + {T_{GI}}}} = \frac{1}{{1 + 2\pi {B^2}{\beta _2}L/N}}.$$

Employing the pre-dispersion-compensation (PDC), the required GI can be halved, and the upper bound of the normalized SE can be estimated as

(12)$$S{E_{PDC - NFDM}} = \frac{{{T_0}}}{{{T_0} + {T_{GI}}/2}} = \frac{1}{{1 + \pi {B^2}{\beta _2}L/N}}.$$

Equation (12) shows that: due to the requirement of the guard interval, the normalized SE of the existing NFDM can’t exceed 1 theoretically. For a given bandwidth and transmission distance system, the only way to increase the normalized SE is to increase the number of modulated nonlinear subcarriers N. However, we can’t increase the number of nonlinear subcarriers infinitely. Because the useful signal duration increases with the increase of the number of subcarriers, which will result in more detrimental signal-noise interaction taking place in the NFT calculation at the receiver [17]. This would degrade system performance dramatically. Here, we propose a nonlinear frequency domain packing strategy and design a brand-new scheme, which can realize more subcarriers multiplexing but the useful signal duration is kept constant. Compared to NFDM system, it can reduce the impact of signal-noise interaction and transmit more information.

4. NFP-NFDM transmission scheme

Firstly, a nonlinear frequency squeeze factor $\alpha$ is introduced in Eq. (8) and name such modulated scheme NFP-NFDM system, which can be expressed as

(13)$$U(\lambda ) = A\sum\limits_{k ={-} (N/2)}^{N/2 - 1} {{s_k}\frac{{\sin (\lambda {T_0}/{T_S} + \alpha k\pi )}}{{\lambda {T_0}/{T_S} + \alpha k\pi }}} ,$$

(14)$$b(\lambda ) = \sqrt {1 - \exp ( - {{|{U(\lambda )} |}^2})} \frac{{U(\lambda )}}{{|{U(\lambda )} |}}.$$

The $\alpha$ is less than 1 and determines the degree of squeeze. When $\alpha$ is equal to 1, the NFP-NFDM will degenerate to conventional NFDM system. It should be noted that the nonlinear frequency in the b-coefficient are not packed in the standard manner but employing the “U-modulation” to achieve the aim of packing indirectly. Figures 1(a)–1(c) show the components distribution of $U(\lambda )$, where the number of nonlinear subcarriers is set to 8. The squeezed b-coefficient is shown in Fig. 1(d). The smaller the squeeze factor, the narrower the $b(\lambda )$. Figures 1(e) and 1(f) compare the time domain waveforms (after INFT) and frequency spectrum separately in different squeeze factor cases. The results show that the signals have the same time duration while their frequency bandwidth are squeezed, which means that the squeezed signal can save the frequency spectrum resource.

Fig. 1. (a-c) The components distribution of $U(\lambda )$ with different squeeze factor, where the curves with different colors denote different components, (d) their corresponding b coefficient, (e) time domain waveforms (after INFT) and (f) (linear) frequency spectrum

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4.1 Process of NFP

In the NFP-NFDM system, two modules are inserted. One is at the DSP of transmitter side and its role is to perform NFP. The other is at the DSP of receiver side and its function is to un-pack and mitigate the inter carrier interference (ICI) induced by the NFP.

The procedure of NFP is shown in Fig. 2. First, the information is encoded in the spectrally efficient frequency division multiplexing (SEFDM) signal. It can be described as

(15)$${\bf X = FS},$$

where S is the $N \times 1$ vector, which is comprised of the symbols ${s_k}$, F is the $N \times N$ operator matrix and ${\bf X}$ is the $N \times 1$ SEFDM signal vector [35]. Then the GI is inserted and the Fourier transform is performed on the SEFDM signal to obtain the $U(\lambda )$. After a conversion from $U(\lambda )$ to $b(\lambda )$, the NFP is completed.

Fig. 2. QAM symbols mapping and the procedure of nonlinear frequency packing.

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As shown in Fig. 3, at the DSP of receiver side, the process is reversed to perform nonlinear frequency un-packing, and the demodulated symbols from the SEFDM signal ${\bf X}$ can be expressed as

(16)$$\textrm{ }{\bf Y = }{{\bf F}^{\bf H}}{\bf X = }{{\bf F}^{\bf H}}{\bf FS = CS},$$

where ${\bf Y}$ is the $N \times 1$ demodulated symbols vector, ${{\bf F}^{\bf H}}$ is the conjugate transpose of matrix ${\bf F}$, and C is the $N \times N$ correlation matrix [35].

Fig. 3. The procedure of nonlinear frequency un-packing, ICI cancellation and symbols de-mapping.

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In general, inverse fractional Fourier transform (IFrFT) can be applied to get the SEFDM signals [28,36], which can be expressed as

(17)$${x_n} = \frac{1}{{\sqrt N }}\sum\limits_{k = 0}^{N - 1} {{s_k}\exp \left( {\frac{{j2\pi k\alpha n}}{N}} \right)} \textrm{ }n = 0,1,2,\ldots N - 1,$$

where ${x_n}$ and ${s_k}$ is the element of SEFDM signal vector ${\bf X}$ and symbol vector ${\bf S}$, respectively. The operator ${\bf F}$ and ${{\bf F}^{\bf H}}$corresponds to IFrFT and fractional Fourier transform (FrFT) operation, respectively. The element of its corresponding correlation matrix is

(18)$${\bf C}_{m,n}^{FrFT} = \left\{ {\begin{array}{cc} 1 & m = n\\ \frac{1}{N}\frac{{1 - \exp [j2\pi \alpha ( - m + n)]}}{{1 - \exp [j2\pi \alpha ( - m + n)/N]}}& \textrm{ }m \ne n \end{array}} \right..$$

When the squeeze factor $\alpha $ is equal to 1, the matrix ${\bf C}_{m,n}^{FrFT}$ degenerates to a unit matrix and ${\bf Y = S}$. The ICI does not exist. When $\alpha $ is less than 1, the correlation matrix ${\bf C}_{}^{FrFT}$ is a complex-valued matrix, and demodulated symbols vector Y is interfered by ICI. It should be noted that the FrFT results in high ICI cancellation complexity due to the complex valued correlation matrix ${\bf C}_{}^{FrFT}$. The detailed reasons will be described in the next section.

To simply the computational complexity of the ICI cancellation algorithm, we propose a modified inverse fractional Fourier transform (MIFrFT), which can be expressed as

(19)$${x_n} = \frac{1}{{\sqrt N }}\sum\limits_{k ={-} N/2}^{N/2 - 1} {{s_k}\exp \left( {\frac{{j2\pi k\alpha ( - N + 1 + 2n)}}{{2N}}} \right)} \textrm{ }n = 0,1,2,\ldots N - 1.$$

The element of the corresponding correlation matrix is

(20)$${\bf C}_{m,n}^{MFrFT} = \left\{ {\begin{array}{cc} 1&m = n\\ \frac{1}{N}\frac{{\sin ({\pi \alpha ( - m + n)} )}}{{\sin ({\pi \alpha ( - m + n)/N} )}}&m \ne n \end{array}} \right..$$

The nature of real value of correlation matrix ${{\bf C}^{MFrFT}}$ stems from the adjustment of initial phase of the subcarriers. Due to the real-valued matrix ${{\bf C}^{MFrFT}}$, the ICI cancellation can be implemented on the real part and imaginary part of the demodulated symbols vector Y separately, which would reduce the computational complexity significantly. In our work, the sphere decoder (SD) is used to perform ICI cancellation and obtain the decision S_SD.

Using the MIFrFT, the NFP-NFDM signal is generated. The detailed process is shown in Fig. 4. Some parameters and function are defined as follows, where R₀ stands for up-sampling factor and $\eta = ({{T_0} + {T_{GI}}} )/{T_0}$ is ratio of total block time to useful time. The $f({\bf V,}M)$ is a function of padding zero at both ends of vector ${\bf V}$, which is defined as

(21)$$f({\bf V,}M) = [{\underbrace{{0,\ldots 0}}_{M},v(0),v(1),\ldots v(n - 1),\underbrace{{0,\ldots 0}}_{M}} ].$$

First, padding zero is performed at both ends of the symbol vector ${\bf S}$ for up-sampling. The MIFrFT is applied to generate the SEFDM signal. Afterwards, the GI is inserted and the NFP $U(\lambda )$ is obtained by DFT operation. The circle shift operation is to move the zero-frequency component to the center of $U(\lambda )$. Then the $U(\lambda )$ is converted to the corresponding scattering coefficient $b(\lambda )$. At last, the NFP-NFDM time domain signal is generated by INFT.

Fig. 4. Flow chart of generating the NFP-NFDM signal

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4.2 ICI impact and cancellation of NFP-NFDM

In the NFP-NFDM system, the absolute values of the elements of correlation matrix ${{\bf C}^{MFrFT}}$ and corresponding symbols constellations with different squeeze factor are shown in Fig. 5, where the number of nonlinear subcarriers is 32 and the impact of noise and impairments from the fiber link are not considered. When the squeeze factor $\alpha $ is equal to 1, there is no ICI. The smaller the squeeze factor $\alpha $, the severer the ICI.

Fig. 5. (a-c). Three-dimensional interference comparison of correlation matrix ${{\bf C}^{MFrFT}}$ with different squeeze factors (the upper list), and the corresponding demodulated QAM constellations interfered by ICI (the lower list), where the number of nonlinear subcarriers is 32.

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To cancel the impact of ICI, the maximum likelihood detection (MLD) was considered [37]. According to the demodulated QAM symbols vector ${\bf Y}$ and the ${{\bf C}^{MFrFT}}$, the final decision can be calculated as

(22)$${{\bf S}_{ML}} = \mathop {\arg \min }\limits_{{\bf S} \in D} {||{{\bf Y} - {{\bf C}^{MFrFT}}{\bf S}} ||^2},$$

where ||.|| denotes the Euclidean norm, and D is the set of all QAM symbol vectors. The complexity of MLD increases exponentially both with the number of subcarriers and the cardinality of modulation format. As the correlation matrix ${{\bf C}^{MFrFT}}$ we constructed is a real valued matrix, and the MLD can be simplified as

(23)$${{\bf S}_{R,ML}} = \mathop {\arg \min }\limits_{{{\bf S}_R} \in {D_R}} {||{{{\bf Y}_R} - {{\bf C}^{MFrFT}}{{\bf S}_R}} ||^2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\bf S}_{I,ML}} = \mathop {\arg \min }\limits_{{{\bf S}_I} \in {D_I}} {||{{{\bf Y}_I} - {{\bf C}^{MFrFT}}{{\bf S}_I}} ||^2},$$

where the subscript R and I denote real part and imaginary part, respectively. ${D_R}$ and ${D_I}$ are the sets of the real part and imaginary part of all QAM symbol vectors, respectively. To further simplify the complexity of MLD, the sphere decoder (SD), initially proposed for MIMO system [38], was applied in our system. Compared to MLD, there is no any performance degradation [35]. The SD algorithm searches the candidates in the N dimension hyperspace sphere with the radius ${\beta _1}$ and${\beta _2}$, which can be expressed as

(24)$${{\bf S}_{R,SD}} = \mathop {\arg \min }\limits_{{{\bf S}_R} \in {D_R}} \{{{{||{{{\bf Y}_R} - {{\bf C}^{MFrFT}}{{\bf S}_R}} ||}^2} < {\beta_{`1}}} \},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\bf S}_{I,SD}} = \mathop {\arg \min }\limits_{{{\bf S}_I} \in {D_I}} \{{{{||{{{\bf Y}_I} - {{\bf C}^{MFrFT}}{{\bf S}_I}} ||}^2} < {\beta_2}} \}.$$

The radius ${\beta _1}$ and ${\beta _2}$ are constantly updated until the symbol vector with the minimum radius is found. The complexity of SD algorithm is dependent to the predefined initial radius, which is estimated by the iterative detection [38]. Figure 6 shows an example of SD algorithm, where 3 nonlinear subcarriers are modulated and the modulation format of subcarriers is pulse-amplitude modulation-2 (2-PAM). The nodes denote the candidate symbols. As shown in Fig. 6, employing the SD, an initial decision is obtained after the third step, and the radius of hyperspace sphere is updated. Then the other nodes within the new sphere space are visited and other paths are tested until finding the path that has the minimum radius. After 10 steps (visited nodes), the final decision is obtained. More detailed description about the SD algorithm can be found in [38].

Fig. 6. An example of SD algorithm (tree structure), where 3 nonlinear subcarriers are modulated and the modulation format is 2-PAM.

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The complexity of ICI cancellation is related to the number of the visited nodes [38]. It should be noted that the MLD need visit all the nodes but the SD only need visit the partial nodes. In the tree structure, the total number of the nodes is equal to $B + {B^2} + {B^3} + \ldots {B^N} = B({1 - {B^N}} )/({1 - B} )$, which is related to the depth (the number of subcarriers N) and the number of the branches per node (B). It should be noted that the branches (B) of the node is related to the method of the NFP. In the case of NFP employing the FrFT, the B is equal to M, where M is the order of modulation format. In the case of NFP employing the MFrFT, the real part and imaginary part of the demodulated symbols vector can be separated to perform the ICI cancellation, and the B is equal to M^1/2(such as 16QAM, B=4). The complexity comparison of ICI cancellation between the NFP employing the FrFT and MFrFT is shown in Table 1.

Table 1. Number of visited nodes in ICI cancellation

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In our work, the MFrFT is employed to perform the NFP and the SD algorithm is used to carry out ICI cancellation. The detailed complexity comparison between MLD and SD in our simulation system is shown in the next section.

5. Numerical results and analysis

In this section, we performed the performance comparison of NFP-NFDM and NFDM through numerical simulation. The schematic diagram of simulation platform and DSP are shown in Fig. 7. At the DSP of transmitter side, using the scheme described in Fig. 4 to generate the NFP-NFDM signal, where Toeplitz Inner Bordering method was applied to carry out INFT [39]. In addition, the PDC is applied to halve the GI. The channel link consists of spans of 12×80 km SSMF with attenuation coefficient α=0.2 dB/km, chromatic coefficient D=16.89 ps/(nm×km) and nonlinear parameter γ=1.3 W^-1km^-1. The split-step Fourier method was applied to simulate the fiber channel and the step length was 1 km. In the fiber link, the signals were amplified by EDFA with 5 dB noise figure and a 1 nm bandwidth optical band pass filter (OBPF) was followed to filter noise. At the receiver side, the normalization operation was performed firstly and the scattering coefficient $b(\lambda )$ was calculated by employing the fast NFT software package [40]. Then half of the channel response was reversed due to the use of the PDC. After the conversion from $b(\lambda )$ to $U(\lambda )$, the nonlinear frequency un-packing was carried out. Compared to NFDM system, an ICI cancelation module was added in NFP-NFDM system. Finally, the Q-factor was derived through direct error counting.

Fig. 7. System platform and the DSP flow of NFP-NFDM and NFDM at transmitter and receiver side

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In the given 32 GHz bandwidth, the number of nonlinear subcarriers was 32 in NFDM system. For NFP-NFDM with the squeeze factor of 0.94, 0.89, 0.84, 0.8, 0.76 and 0.74, the number of the nonlinear subcarriers were 34, 36, 38, 40, 42 and 44, respectively. The modulation format of NFDM and NFP-NFDM systems were both 16QAM.The effective signal duration and GI of NFDM and NFP-NFDM were 1 ns and 2 ns, respectively. The up-sampling factor was R₀=4. The samples of the signal block in time domain and nonlinear domain were same and equaled to (R₀×number of nonlinear subcarriers × ratio of total block time to useful time) samples per block. The linear spectrum of NFDM and NFP-NFDM with different squeeze factor are shown in Fig. 8. The occupied bandwidth of NFDM and NFP-NFDM is same, where the launch power of signal is -1dBm.

Fig. 8. Linear frequency spectrum of NFDM (α=1) and NFP-NFDM (α=0.94, 0.89, 0.84, 0.8,0.76, 0.73), where the launch power is -1dBm.

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We first performed a back to back (BTB) simulation comparison. The noise was added to the signal directly, i.e. additive white Gaussian noise (AWGN) scenario. We try to add the same noise power with that of next 960 km fiber transmission scenario, while the BER of both NFDM and NFP-NFDM are nearly zero at the optimal power range. To observe the performance differences clearly between NFDM and NFP-NFDM, the noise power in BTB scenario was increased and 3 dB higher than that of transmission scenario. As shown in Fig. 9, the NFP-NFDM with smaller squeeze factor behaves worse performance. But at the optimal launch power, the performance of NFDM and NFP-NFDM ($\alpha = 0.94,{\kern 1pt} {\kern 1pt} 0.89, {\kern 1pt} {\kern 1pt} 0.84$) is almost same, which shows the possibility of high SE NFP-NFDM transmission.

Fig. 9. Q-factor as a function of launch power (BTB) for NFDM(α=1) and NFP-NFDM (α=0.94, 0.89, 0.84, 0.8, 0.76 and 0.73)

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Next, the 960 km fiber transmission of NFDM and NFP-NFDM signal was simulated. As shown in Fig. 10, the launch power is swept from -17dBm to -1.5dBm. Interestingly, for around the optimal launch power case, the NFP-NFDM (α=0.94) behaves best performance and offers 1.1 dB performance gain than unpacked NFDM system (but with ICI cancellation). That is an interesting phenomenon, it may imply that signal-noise interaction can be partially mitigated by introducing the known ICI. Remind that the SD algorithm is only for known ICI rather than a blind compensation, and it will not change the signal quality if $\alpha = 1$ because its corresponding correlation matrix ${C^{MFrFT}}$ will degenerate to a unit matrix. Therefore, the most likely reason for this outperformance is that packed subcarriers get less impact when signal evolves in the fiber mixed with ASE noise. The Q-factor with 0.89 squeeze factor is similar to the unpacked one at optimal launch power, while the SE can be increased by 12.5% compared with unpacked system. These results show that NFP scheme may be more promising than the OFDM-like spectrum approach.

Fig. 10. Q-factor as a function of launch power after 960 km transmission for NFDM(α=1) and NFP-NFDM (α=0.94, 0.89, 0.84, 0.8, 0.76 and 0.73).

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When the squeeze factor continues to decrease, the performance becomes worse. And the BER of NFP-NFDM with the squeeze factor of $\alpha = 0.76,0.73$ cannot reach the hard decision-FEC threshold (HD-FEC, 3.8×10⁻³ threshold) [41,42]. In addition, for the low launch power cases, the NFDM behaves better performance than NFP-NFDM, and the trend of the curves is same with the AWGN scenario, which show that the ASE noise in NFP-NFDM system can be approximately regarded as the AWGN model for low launch power cases.

Assuming an ideal HD-FEC code, the achievable information rate (AIR) can be estimated through pre-FEC BER

(25)$$AIR = {R_B} \cdot {\log _2}M \cdot {R_C},$$

where ${R_B}$ is baud rate, M is the order of modulation format. And ${R_C}$ is coding rate, which is equal to${\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R_C} = 1 + BER \cdot {\log _2}BER + (1 - BER) \cdot {\log _2}(1 - BER)$ [35]. The AIR of NFDM and NFP-NFDM with BER less than 3.8e-3 are calculated. In Fig. 11, the AIR of NFDM and NFP-NFDM are compared. For NFDM, the maximum AIR is 42.2 Gbps (1/3ns×32× 4×0.988 bit) and the its corresponding normalized SE is 0.329 (AIR/log₂M/32 GHz) symbol/s/Hz. The maximum AIR and normalized SE for different squeeze factors NFP-NFDM system are shown in Table 2. Compared to NFDM, the SE of NFP-NFDM system with different squeeze factor are increased by 7.2%, 12.8%, 17.8%, 22.9%, respectively.

Fig. 11. AIR as a function of launch power after 960 km transmission for NFDM(α=1) and NFP-NFDM (α=0.94, 0.89, 0.84, 0.8).

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Table 2. Maximum AIR and Normalized SE for different squeeze factors NFP-NFDM

View Table | View all tables in this article

The complexity comparison between MLD and SD is shown in Fig. 12. Compared to MLD, the complexity of SD is reduced significantly. However, with the decrease of the squeeze factor, the complexity of SD is also increased. So, the ICI cancellation algorithm with lower complexity is the key to achieving the higher SE NFP-NFDM system.

Fig. 12. Number of visited nodes as a function of squeeze factor at the optimal performance after 960 km transmission for NFP-NFDM (α=0.94, 0.89, 0.84, 0.8).

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6. Further discussions and perspectives

Compared with NFDM, the spacing of the nonlinear subcarriers in NFP-NFDM is squeezed and more nonlinear subcarriers can be packed in the same bandwidth and block time. The upper bound of the normalized SE can be estimated as

(26)$$S{E_{NFP - NFDM,PDC}} = \frac{{{N_{NFP - NFDM}}}}{{({T_0} + {T_{GI}})B}} = \frac{{N/\alpha }}{{({T_0} + {T_{GI}})B}} = \frac{1}{\alpha }S{E_{PDC - NFDM}},$$

where the PDC is employed. When the number of nonlinear subcarriers is N in NFDM system, the number of corresponding subcarriers is ${N_{NFP - NFDM}} = N/\alpha$ for the NFP-NFDM system with the squeeze factor $\alpha $. Compared to NFDM system, the upper bound of normalized SE can be increased by $(S{E_{NFP - NFDM,PDC}} - S{E_{PDC - NFDM}})/S{E_{PDC - NFDM}} = ( 1 - \alpha ) /\alpha $. Figure 13 shows the upper bound of the normalized SE for NFDM and NFP-NFDM ($\alpha = 0.8$). The upper bound of SE in NFP-NFDM can exceed 1 and equal to $1/\alpha $, theoretically. However, when the number of subcarriers is increased, the complexity of ICI cancellation will be increased as well. In addition, in NFP-NFDM system with the smaller squeeze factor, the ICI becomes more serious and the complexity of ICI cancellation will be increased. The efficient ICI cancellation algorithm will be explored in our future work. In addition, it should be noted that even with the best ICI cancellation algorithm, the upper bound of the SE curve cannot be reached due to the interaction of noise. In this work, due to the limit of ICI cancellation algorithm complexity, the SE of NFP-NFDM system is not more than that of current demonstrated NFDM system. But we highlight the potential of the NFP-NFDM transmission and illustrate the possibility of a new packing approach in NFD to increase the SE.

Fig. 13. Normalized SE of NFDM and NFP-NFDM ($\alpha = 0.8$) with PDC as a function of the number of nonlinear subcarriers, where the transmission distance is 960 km and the bandwidth of NFDM and NFP-NFDM is 32 GHz.

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7. Conclusion

To the best of our knowledge, we first propose the nonlinear frequency packing (NFP)-NFDM system. The NFP-NFDM system can add more subcarriers and then realize higher SE than NFDM system in the fixed linear bandwidth and block time. The NFP introduces the known inter-carrier interference (ICI) to the system, which can be compensated by DSP algorithm at receiver side. In this work, we proposed a modified fractional Fourier transform (MFrFT) to perform the NFP and the complexity of ICI cancellation is reduced significantly. However, the complexity of ICI cancellation is still dependent on the number of subcarriers and squeeze factor. The lower complexity ICI cancellation algorithm will become the key factor of NFP-NFDM. The numerical simulation setup is built for the performance test. Back-to-back test shows that the known ICI can be efficiently compensated when the squeeze factor is not small. In fiber transmission scenario, the NFP-NFDM with 0.94 squeeze factor shows 1.1 dB performance gain and offers 7.2% increase in the SE compared with the unpacked-NFDM system. This implies that nonorthogonal signaling can bring the advantages on signal-noise interaction reduction. And NFP may be a better candidate for data loading method in NFDM system. The performance of NFP-NFDM with 0.89 squeeze factor is almost same with NFDM and offers 12.8% SE increasing. The squeeze factor can be lower up to 0.8, and the corresponding Q-factor reaches the HD-FEC (3.8e-3) threshold at the optimal launch power. In that case, NFP-NFDM offers 22.9% increase in the SE. The upper bound of the normalized SE in NFP-NFDM system is also derived and it can break through 1 naturally, depends on the used squeeze factor. This work offers a new direction to increase the SE and paved the way for high SE NFDM system.

Funding

State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) (IPOC2019ZZ02); Beijing Excellent Ph.D. Thesis Guidance Foundation (CX2018113).

Disclosures

The authors declare no conflicts of interest.

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	NFDM (α=1)	NFP-NFDM (α=0.94)	NFP-NFDM (α=0.89)	NFP-NFDM (α=0.84)	NFP-NFDM (α=0.8)
AIR (Gbps)	42.2	45.2	47.6	49.7	51.8
Normalized SE (symbol/s/Hz)	0.329	0.353	0.372	0.388	0.405

	NFDM (α=1)	NFP-NFDM (α=0.94)	NFP-NFDM (α=0.89)	NFP-NFDM (α=0.84)	NFP-NFDM (α=0.8)
AIR (Gbps)	42.2	45.2	47.6	49.7	51.8
Normalized SE (symbol/s/Hz)	0.329	0.353	0.372	0.388	0.405

Nonlinear-frequency-packing nonlinear frequency division multiplexing transmission

Abstract

1. Introduction

2. Channel model and nonlinear Fourier transform

3. NFDM transmissions with b-modulation and the strategy of increasing SE

4. NFP-NFDM transmission scheme

4.1 Process of NFP

4.2 ICI impact and cancellation of NFP-NFDM

5. Numerical results and analysis

6. Further discussions and perspectives

7. Conclusion

Funding

Disclosures

References

Cited By

Figures (13)

Tables (2)

Equations (26)

Optics Express

	MLD	SD
NFP employing the FrFT	$M (1 - M^{N}) / (1 - M)$	Less than $M (1 - M^{N}) / (1 - M)$
NFP employing the MFrFT	$2 M^{1 / 2} (1 - M^{N / 2}) / (1 - M^{1 / 2})$	Less than $2 M^{1 / 2} (1 - M^{N / 2}) / (1 - M^{1 / 2})$